P. 148 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14701-14800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	psr0cl 14701*	The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- .0. = ( 0g ` R )   &    |- B = ( Base ` S )   =>    |- ( ph -> ( D X. { .0. } ) e. B )
Theorem	psr0lid 14702*	The zero element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- .0. = ( 0g ` R )   &    |- B = ( Base ` S )   &    |- .+ = ( +g ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( ( D X. { .0. } ) .+ X ) = X )
Theorem	psrnegcl 14703*	The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- N = ( invg ` R )   &    |- B = ( Base ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( N o. X ) e. B )
Theorem	psrlinv 14704*	The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- N = ( invg ` R )   &    |- B = ( Base ` S )   &    |- ( ph -> X e. B )   &    |- .0. = ( 0g ` R )   &    |- .+ = ( +g ` S )   =>    |- ( ph -> ( ( N o. X ) .+ X ) = ( D X. { .0. } ) )
Theorem	psrgrp 14705	The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof shortened by SN, 7-Feb-2025.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> S e. Grp )
Theorem	psr0 14706*	The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- O = ( 0g ` R )   &    |- .0. = ( 0g ` S )   =>    |- ( ph -> .0. = ( D X. { O } ) )
Theorem	psrneg 14707*	The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- N = ( invg ` R )   &    |- B = ( Base ` S )   &    |- M = ( invg ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( M ` X ) = ( N o. X ) )
Theorem	psr1clfi 14708*	The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Ring )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- .0. = ( 0g ` R )   &    |- .1. = ( 1r ` R )   &    |- U = ( x e. D |-> if ( x = ( I X. { 0 } ) , .1. , .0. ) )   &    |- B = ( Base ` S )   =>    |- ( ph -> U e. B )
Theorem	reldmmpl 14709	The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- Rel dom mPoly
Theorem	mplvalcoe 14710*	Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = { f e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) }   =>    |- ( ( I e. V /\ R e. W ) -> P = ( S ↾s U ) )
Theorem	mplbascoe 14711*	Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = ( Base ` P )   =>    |- ( ( I e. V /\ R e. W ) -> U = { f e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) } )
Theorem	mplelbascoe 14712*	Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = ( Base ` P )   =>    |- ( ( I e. V /\ R e. W ) -> ( X e. U <-> ( X e. B /\ E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( X ` b ) = .0. ) ) ) )
Theorem	fnmpl 14713	mPoly has universal domain. (Contributed by Jim Kingdon, 5-Nov-2025.)
|- mPoly Fn ( _V X. _V )
Theorem	mplrcl 14714	Reverse closure for the polynomial index set. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
|- P = ( I mPoly R )   &    |- B = ( Base ` P )   =>    |- ( X e. B -> I e. _V )
Theorem	mplval2g 14715	Self-referential expression for the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- U = ( Base ` P )   =>    |- ( ( I e. V /\ R e. W ) -> P = ( S ↾s U ) )
Theorem	mplbasss 14716	The set of polynomials is a subset of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- U = ( Base ` P )   &    |- B = ( Base ` S )   =>    |- U C_ B
Theorem	mplelf 14717*	A polynomial is defined as a function on the coefficients. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- P = ( I mPoly R )   &    |- K = ( Base ` R )   &    |- B = ( Base ` P )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- ( ph -> X e. B )   =>    |- ( ph -> X : D --> K )
Theorem	mplsubgfilemm 14718*	Lemma for mplsubgfi 14721. There exists a polynomial. (Contributed by Jim Kingdon, 21-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> E. j j e. U )
Theorem	mplsubgfilemcl 14719	Lemma for mplsubgfi 14721. The sum of two polynomials is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- .+ = ( +g ` S )   =>    |- ( ph -> ( X .+ Y ) e. U )
Theorem	mplsubgfileminv 14720	Lemma for mplsubgfi 14721. The additive inverse of a polynomial is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. U )   &    |- N = ( invg ` S )   =>    |- ( ph -> ( N ` X ) e. U )
Theorem	mplsubgfi 14721	The set of polynomials is closed under addition, i.e. it is a subgroup of the set of power series. (Contributed by Mario Carneiro, 8-Jan-2015.) (Proof shortened by AV, 16-Jul-2019.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> U e. (SubGrp ` S ) )
Theorem	mpl0fi 14722*	The zero polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
|- P = ( I mPoly R )   &    |- O = ( 0g ` R )   &    |- .0. = ( 0g ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> .0. = ( x e. ( NN0 ^m I ) |-> O ) )
Theorem	mplplusgg 14723	Value of addition in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- Y = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- .+ = ( +g ` Y )   =>    |- ( ( I e. V /\ R e. W ) -> .+ = ( +g ` S ) )
Theorem	mpladd 14724	The addition operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- P = ( I mPoly R )   &    |- B = ( Base ` P )   &    |- .+ = ( +g ` R )   &    |- .+b = ( +g ` P )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .+b Y ) = ( X oF .+ Y ) )
Theorem	mplnegfi 14725	The negative function on multivariate polynomials. (Contributed by SN, 25-May-2024.)
|- P = ( I mPoly R )   &    |- B = ( Base ` P )   &    |- N = ( invg ` R )   &    |- M = ( invg ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( M ` X ) = ( N o. X ) )
Theorem	mplgrpfi 14726	The polynomial ring is a group. (Contributed by Mario Carneiro, 9-Jan-2015.)
|- P = ( I mPoly R )   =>    |- ( ( I e. Fin /\ R e. Grp ) -> P e. Grp )
PART 9  BASIC TOPOLOGY
9.1  Topology
9.1.1  Topological spaces A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union), and it may sometimes be more convenient to consider topologies without reference to the underlying set.
9.1.1.1  Topologies
Syntax	ctop 14727	Syntax for the class of topologies.
class Top
Definition	df-top 14728*	Define the class of topologies. It is a proper class. See istopg 14729 and istopfin 14730 for the corresponding characterizations, using respectively binary intersections like in this definition and nonempty finite intersections. The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241. (Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)
|- Top = { x | ( A. y e. ~P x U. y e. x /\ A. y e. x A. z e. x ( y i^i z ) e. x ) }
Theorem	istopg 14729*	Express the predicate " J is a topology". See istopfin 14730 for another characterization using nonempty finite intersections instead of binary intersections. Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use T to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- ( J e. A -> ( J e. Top <-> ( A. x ( x C_ J -> U. x e. J ) /\ A. x e. J A. y e. J ( x i^i y ) e. J ) ) )
Theorem	istopfin 14730*	Express the predicate " J is a topology" using nonempty finite intersections instead of binary intersections as in istopg 14729. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.)
|- ( J e. Top -> ( A. x ( x C_ J -> U. x e. J ) /\ A. x ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) -> |^| x e. J ) ) )
Theorem	uniopn 14731	The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
|- ( ( J e. Top /\ A C_ J ) -> U. A e. J )
Theorem	iunopn 14732*	The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
|- ( ( J e. Top /\ A. x e. A B e. J ) -> U_ x e. A B e. J )
Theorem	inopn 14733	The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
|- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A i^i B ) e. J )
Theorem	fiinopn 14734	The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
|- ( J e. Top -> ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> |^| A e. J ) )
Theorem	unopn 14735	The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A u. B ) e. J )
Theorem	0opn 14736	The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
|- ( J e. Top -> (/) e. J )
Theorem	0ntop 14737	The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
|- -. (/) e. Top
Theorem	topopn 14738	The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
|- X = U. J   =>    |- ( J e. Top -> X e. J )
Theorem	eltopss 14739	A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
|- X = U. J   =>    |- ( ( J e. Top /\ A e. J ) -> A C_ X )
9.1.1.2  Topologies on sets
Syntax	ctopon 14740	Syntax for the function of topologies on sets.
class TopOn
Definition	df-topon 14741*	Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- TopOn = ( b e. _V |-> { j e. Top | b = U. j } )
Theorem	funtopon 14742	The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
|- Fun TopOn
Theorem	istopon 14743	Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- ( J e. (TopOn ` B ) <-> ( J e. Top /\ B = U. J ) )
Theorem	topontop 14744	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( J e. (TopOn ` B ) -> J e. Top )
Theorem	toponuni 14745	The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( J e. (TopOn ` B ) -> B = U. J )
Theorem	topontopi 14746	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- J e. (TopOn ` B )   =>    |- J e. Top
Theorem	toponunii 14747	The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- J e. (TopOn ` B )   =>    |- B = U. J
Theorem	toptopon 14748	Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- X = U. J   =>    |- ( J e. Top <-> J e. (TopOn ` X ) )
Theorem	toptopon2 14749	A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
|- ( J e. Top <-> J e. (TopOn ` U. J ) )
Theorem	topontopon 14750	A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
|- ( J e. (TopOn ` X ) -> J e. (TopOn ` U. J ) )
Theorem	toponrestid 14751	Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.)
|- A e. (TopOn ` B )   =>    |- A = ( A ↾t B )
Theorem	toponsspwpwg 14752	The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
|- ( A e. V -> (TopOn ` A ) C_ ~P ~P A )
Theorem	dmtopon 14753	The domain of TopOn is _V. (Contributed by BJ, 29-Apr-2021.)
|- dom TopOn = _V
Theorem	fntopon 14754	The class TopOn is a function with domain _V. (Contributed by BJ, 29-Apr-2021.)
|- TopOn Fn _V
Theorem	toponmax 14755	The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( J e. (TopOn ` B ) -> B e. J )
Theorem	toponss 14756	A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ A e. J ) -> A C_ X )
Theorem	toponcom 14757	If K is a topology on the base set of topology J, then J is a topology on the base of K. (Contributed by Mario Carneiro, 22-Aug-2015.)
|- ( ( J e. Top /\ K e. (TopOn ` U. J ) ) -> J e. (TopOn ` U. K ) )
Theorem	toponcomb 14758	Biconditional form of toponcom 14757. (Contributed by BJ, 5-Dec-2021.)
|- ( ( J e. Top /\ K e. Top ) -> ( J e. (TopOn ` U. K ) <-> K e. (TopOn ` U. J ) ) )
Theorem	topgele 14759	The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
|- ( J e. (TopOn ` X ) -> ( { (/) , X } C_ J /\ J C_ ~P X ) )
9.1.1.3  Topological spaces
Syntax	ctps 14760	Syntax for the class of topological spaces.
class TopSp
Definition	df-topsp 14761	Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
|- TopSp = { f | ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) }
Theorem	istps 14762	Express the predicate "is a topological space". (Contributed by Mario Carneiro, 13-Aug-2015.)
|- A = ( Base ` K )   &    |- J = ( TopOpen ` K )   =>    |- ( K e. TopSp <-> J e. (TopOn ` A ) )
Theorem	istps2 14763	Express the predicate "is a topological space". (Contributed by NM, 20-Oct-2012.)
|- A = ( Base ` K )   &    |- J = ( TopOpen ` K )   =>    |- ( K e. TopSp <-> ( J e. Top /\ A = U. J ) )
Theorem	tpsuni 14764	The base set of a topological space. (Contributed by FL, 27-Jun-2014.)
|- A = ( Base ` K )   &    |- J = ( TopOpen ` K )   =>    |- ( K e. TopSp -> A = U. J )
Theorem	tpstop 14765	The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)
|- J = ( TopOpen ` K )   =>    |- ( K e. TopSp -> J e. Top )
Theorem	tpspropd 14766	A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- ( ph -> ( Base ` K ) = ( Base ` L ) )   &    |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )   =>    |- ( ph -> ( K e. TopSp <-> L e. TopSp ) )
Theorem	topontopn 14767	Express the predicate "is a topological space". (Contributed by Mario Carneiro, 13-Aug-2015.)
|- A = ( Base ` K )   &    |- J = (TopSet ` K )   =>    |- ( J e. (TopOn ` A ) -> J = ( TopOpen ` K ) )
Theorem	tsettps 14768	If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)
|- A = ( Base ` K )   &    |- J = (TopSet ` K )   =>    |- ( J e. (TopOn ` A ) -> K e. TopSp )
Theorem	istpsi 14769	Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)
|- ( Base ` K ) = A   &    |- ( TopOpen ` K ) = J   &    |- A = U. J   &    |- J e. Top   =>    |- K e. TopSp
Theorem	eltpsg 14770	Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
|- K = { <. ( Base ` ndx ) , A >. , <. (TopSet ` ndx ) , J >. }   =>    |- ( J e. (TopOn ` A ) -> K e. TopSp )
Theorem	eltpsi 14771	Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- K = { <. ( Base ` ndx ) , A >. , <. (TopSet ` ndx ) , J >. }   &    |- A = U. J   &    |- J e. Top   =>    |- K e. TopSp
9.1.2  Topological bases
Syntax	ctb 14772	Syntax for the class of topological bases.
class TopBases
Definition	df-bases 14773*	Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 14775). Note that "bases" is the plural of "basis". (Contributed by NM, 17-Jul-2006.)
|- TopBases = { x | A. y e. x A. z e. x ( y i^i z ) C_ U. ( x i^i ~P ( y i^i z ) ) }
Theorem	isbasisg 14774*	Express the predicate "the set B is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
|- ( B e. C -> ( B e. TopBases <-> A. x e. B A. y e. B ( x i^i y ) C_ U. ( B i^i ~P ( x i^i y ) ) ) )
Theorem	isbasis2g 14775*	Express the predicate "the set B is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
|- ( B e. C -> ( B e. TopBases <-> A. x e. B A. y e. B A. z e. ( x i^i y ) E. w e. B ( z e. w /\ w C_ ( x i^i y ) ) ) )
Theorem	isbasis3g 14776*	Express the predicate "the set B is a basis for a topology". Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.)
|- ( B e. C -> ( B e. TopBases <-> ( A. x e. B x C_ U. B /\ A. x e. U. B E. y e. B x e. y /\ A. x e. B A. y e. B A. z e. ( x i^i y ) E. w e. B ( z e. w /\ w C_ ( x i^i y ) ) ) ) )
Theorem	basis1 14777	Property of a basis. (Contributed by NM, 16-Jul-2006.)
|- ( ( B e. TopBases /\ C e. B /\ D e. B ) -> ( C i^i D ) C_ U. ( B i^i ~P ( C i^i D ) ) )
Theorem	basis2 14778*	Property of a basis. (Contributed by NM, 17-Jul-2006.)
|- ( ( ( B e. TopBases /\ C e. B ) /\ ( D e. B /\ A e. ( C i^i D ) ) ) -> E. x e. B ( A e. x /\ x C_ ( C i^i D ) ) )
Theorem	fiinbas 14779*	If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( B e. C /\ A. x e. B A. y e. B ( x i^i y ) e. B ) -> B e. TopBases )
Theorem	baspartn 14780*	A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( ( P e. V /\ A. x e. P A. y e. P ( x = y \/ ( x i^i y ) = (/) ) ) -> P e. TopBases )
Theorem	tgval2 14781*	Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 14794) that ( topGen ` B ) is indeed a topology (on U. B, see unitg 14792). See also tgval 13350 and tgval3 14788. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> ( topGen ` B ) = { x | ( x C_ U. B /\ A. y e. x E. z e. B ( y e. z /\ z C_ x ) ) } )
Theorem	eltg 14782	Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> ( A e. ( topGen ` B ) <-> A C_ U. ( B i^i ~P A ) ) )
Theorem	eltg2 14783*	Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> ( A e. ( topGen ` B ) <-> ( A C_ U. B /\ A. x e. A E. y e. B ( x e. y /\ y C_ A ) ) ) )
Theorem	eltg2b 14784*	Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> ( A e. ( topGen ` B ) <-> A. x e. A E. y e. B ( x e. y /\ y C_ A ) ) )
Theorem	eltg4i 14785	An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( A e. ( topGen ` B ) -> A = U. ( B i^i ~P A ) )
Theorem	eltg3i 14786	The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
|- ( ( B e. V /\ A C_ B ) -> U. A e. ( topGen ` B ) )
Theorem	eltg3 14787*	Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.)
|- ( B e. V -> ( A e. ( topGen ` B ) <-> E. x ( x C_ B /\ A = U. x ) ) )
Theorem	tgval3 14788*	Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 13350 and tgval2 14781. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
|- ( B e. V -> ( topGen ` B ) = { x | E. y ( y C_ B /\ x = U. y ) } )
Theorem	tg1 14789	Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
|- ( A e. ( topGen ` B ) -> A C_ U. B )
Theorem	tg2 14790*	Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
|- ( ( A e. ( topGen ` B ) /\ C e. A ) -> E. x e. B ( C e. x /\ x C_ A ) )
Theorem	bastg 14791	A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> B C_ ( topGen ` B ) )
Theorem	unitg 14792	The topology generated by a basis B is a topology on U. B. Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class TopBases completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.)
|- ( B e. V -> U. ( topGen ` B ) = U. B )
Theorem	tgss 14793	Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
|- ( ( C e. V /\ B C_ C ) -> ( topGen ` B ) C_ ( topGen ` C ) )
Theorem	tgcl 14794	Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)
|- ( B e. TopBases -> ( topGen ` B ) e. Top )
Theorem	tgclb 14795	The property tgcl 14794 can be reversed: if the topology generated by B is actually a topology, then B must be a topological basis. This yields an alternative definition of TopBases. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- ( B e. TopBases <-> ( topGen ` B ) e. Top )
Theorem	tgtopon 14796	A basis generates a topology on U. B. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( B e. TopBases -> ( topGen ` B ) e. (TopOn ` U. B ) )
Theorem	topbas 14797	A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
|- ( J e. Top -> J e. TopBases )
Theorem	tgtop 14798	A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
|- ( J e. Top -> ( topGen ` J ) = J )
Theorem	eltop 14799	Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
|- ( J e. Top -> ( A e. J <-> A C_ U. ( J i^i ~P A ) ) )
Theorem	eltop2 14800*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
|- ( J e. Top -> ( A e. J <-> A. x e. A E. y e. J ( x e. y /\ y C_ A ) ) )